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In this article, we investigate certain asymptotic optimality properties of avery broad class of one-group continuous shrinkage priors for simultaneousestimation and testing of a sparse normal mean vector. Asymptotic optimality ofBayes estimates and posterior concentration properties corresponding to thegeneral class of one-group priors under consideration are studied where thedata is assumed to be generated according to a multivariate normal distributionwith a fixed unknown mean vector. Under the assumption that the number ofnon-zero means is known, we show that Bayes estimators arising out of thisgeneral class of shrinkage priors under study, attain the minimax risk, up tosome multiplicative constant, under the $l_2$ norm. In particular, it is shownthat for the horseshoe-type priors such as the three parameter beta normalmixtures with parameters $a=0.5, b>0$ and the generalized double Pareto priorwith shape parameter $\alpha=1$, the corresponding Bayes estimates becomeasymptotically minimax. Moreover, posterior distributions arising out of thisgeneral class of one-group priors are shown to contract around the true meanvector at the minimax $l_2$ rate for a wide range of values of the globalshrinkage parameter depending on the proportion of non-zero components of theunderlying mean vector. An important and remarkable fact that emerges as aconsequence of one key result essential for proving the aforesaid minimaxityresult is that, within the asymptotic framework of Bogdan et al. (2011), thenatural thresholding rules due to Carvalho et al. (2010) based on thehorseshoe-type priors, asymptotically attain the optimal Bayes risk w.r.t. a$0-1$ loss, up to the correct multiplicative constant and are thus,asymptotically Bayes optimal under sparsity (ABOS).

Asymptotic Minimaxity, Optimal Posterior Concentration and Asymptotic Bayes Optimality of Horseshoe-type Priors Under Sparsity